Package ‘RConics’ - The Comprehensive R Archive Network · Package ‘RConics’ February 19,...
Transcript of Package ‘RConics’ - The Comprehensive R Archive Network · Package ‘RConics’ February 19,...
Package ‘RConics’February 19, 2015
Type Package
Title Computations on Conics
Version 1.0
Date 2014-02-21
Author Emanuel Huber
Maintainer Emanuel Huber <[email protected]>
Description Solve some conic related problems (intersection of conics with lines and con-ics, arc length of an ellipse, polar lines, etc.).
License GPL (>= 2)
Encoding UTF-8
NeedsCompilation no
Repository CRAN
Date/Publication 2014-12-26 14:36:49
R topics documented:RConics-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2(i,j)-cofactor and (i,j)-minor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3addLine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Affine planar transformations matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5arcLengthEllipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6colinear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8conicMatrixToEllipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9conicThrough5Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12ellipseToConicMatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Intersection with conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Join, meet and parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15pEllipticInt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16polar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1
2 RConics-package
quadraticFormToMatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18skewSymmetricMatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19splitDegenerateConic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Index 21
RConics-package Computations on conics
Description
Solve some conic related problems (intersection of conics with lines and conics, arc length of anellipse, polar lines, etc.).
Details
Package: RConicsType: PackageVersion: 1.0Date: 2014-02-21License: GPL 2
Note
Some of the functions are based on the projective geometry. In projective geometry parallel linesmeet at an infinite point and all infinite points are incident to a line at infinity. Points and lines of aprojective plane are represented by homogeneous coordinates, that means by 3D vectors: (x, y, z)for the points and (a, b, c) such that ax+ by+ c = 0 for the lines. The Euclidian points correspondto (x, y, 1), the infinite points to (x, y, 0), the Euclidian lines to (a, b, c) with a 6= 0 or b 6= 0, theline at infinity to (0, 0, 1).
Advice: to plot conics use the package conics from Bernard Desgraupes.
This work was funded by the Swiss National Science Foundation within the ENSEMBLE project(grant no. CRSI_132249).
Author(s)
Emanuel Huber
Maintainer: Emanuel Huber <[email protected]>
References
Richter-Gebert, Jürgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Realand Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4
(i,j)-cofactor and (i,j)-minor 3
(i,j)-cofactor and (i,j)-minor
(i, j)-cofactor and (i, j)-minor of a matrix
Description
Compute the (i, j)-cofactor, respectively the (i, j)-minor of the matrix A. The (i, j)-cofactor isobtained by multiplying the (i, j)-minor by (−1)i+j . The (i, j)-minor of A, is the determinant ofthe (n− 1)× (n− 1) matrix that results by deleting the i-th row and the j-th column of A.
Usage
cofactor(A, i, j)minor(A, i, j)
Arguments
A a square matrix.
i the i-th row.
j the j-th column.
Value
The (i, j)-minor/cofactor of the matrix A (single value).
See Also
adjoint
Examples
A <- matrix(c(1,4,5,3,7,2,2,8,3),nrow=3,ncol=3)Aminor(A,2,3)cofactor(A,2,3)
4 adjoint
addLine Add a "homogeneous" line to a plot.
Description
Add a homogeneous line to a plot. The line parameters must be in homogeneous coordinates, e.g.(a, b, c).
Usage
addLine(l, ...)
Arguments
l a 3× 1 vector of the homogeneous representation of a line.
... graphical parameters such as col, lty and lwd.
Details
addLine is based on abline.
Examples
# two points in homogeneous coordinatesp1 <- c(3,1,1)p2 <- c(0,2,1)
# homogeneous line joining p1 and p2l_12 <- join(p1,p2)l_12
# plotplot(0,0,type="n", xlim=c(-2,5),ylim=c(-2,5),asp=1)points(t(p1))points(t(p2))addLine(l_12,col="red",lwd=2)
adjoint Adjoint matrix
Description
Compute the classical adjoint (also called adjugate) of a square matrix. The adjoint is the transposeof the cofactor matrix.
Affine planar transformations matrix 5
Usage
adjoint(A)
Arguments
A a square matrix.
Value
The adjoint matrix of A (square matrix with the same dimension as A).
See Also
cofactor, minor
Examples
A <- matrix(c(1,4,5,3,7,2,2,8,3),nrow=3,ncol=3)AB <- adjoint(A)B
Affine planar transformations matrix
Affine planar transformation matrix
Description
(3× 3) affine planar transformation matrix corresponding to reflection, rotation, scaling and trans-lation in projective geometry. To transform a point p multiply the transformation matrix A with thehomogeneous coordinates (x, y, z) of p (e.g. ptransformed = Ap).
Usage
reflection(alpha)rotation(theta, pt=NULL)scaling(s)translation(v)
Arguments
alpha the angle made by the line of reflection (in radian).
theta the angle of the rotation (in radian).
pt the homogeneous coordinates of the rotation center (optional).
s the (2× 1) scaling vector in direction x and y.
v the (2× 1) translation vector in direction x and y.
6 arcLengthEllipse
Value
A (3× 3) affine transformation matrix.
References
Richter-Gebert, Jürgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Realand Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4
Examples
p1 <- c(2,5,1) # homogeneous coordinate
# rotationr_p1 <- rotation(4.5) %*% p1
# rotation centered in (3,1)rt_p1 <- rotation(4.5, pt=c(3,1,1)) %*% p1
# translationt_p1 <- translation(c(2,-4)) %*% p1
# scalings_p1 <- scaling(c(-3,1)) %*% p1
# plotplot(t(p1),xlab="x",ylab="y", xlim=c(-5,5),ylim=c(-5,5),asp=1)abline(v=0,h=0, col="grey",lty=1)abline(v=3,h=1, col="grey",lty=3)points(3,1,pch=4)points(t(r_p1),col="red",pch=20)points(t(rt_p1),col="blue",pch=20)points(t(t_p1),col="green",pch=20)points(t(s_p1),col="black",pch=20)
arcLengthEllipse Arc length of an ellipse
Description
This function computes the arc length of an ellipse centered in (0, 0) with the semi-axes alignedwith the x- and y-axes. The arc length is defined by the points 1 and 2. These two points do notneed to lie exactly on the ellipse: the x-coordinate of the points and the quadrant where they liedefine the positions on the ellipse used to compute the arc length.
Usage
arcLengthEllipse(p1, p2 = NULL, saxes, n = 5)
arcLengthEllipse 7
Arguments
p1 a (2× 1) vector of the Cartesian coordinates of point 1.
p2 a (2× 1) vector of the Cartesian coordinates of point 2 (optional).
saxes a (2× 1) vector of length of the semi-axes of the ellipse.
n the number of iterations used in the numerical approximation of the incompleteelliptic integral of the second kind.
Details
If the coordinates p2 of the point 2 are omitted the function arcLengthEllipse computes the arclength between the point 1 and the point defined by (0, b), b beeing the minor semi-axis.
Value
The length of the shortest arc of the ellipse defined by the points 1 and 2.
References
Van de Vel, H. (1969). On the series expansion method for Computing incomplete elliptic integralsof the first and second kinds, Math. Comp. 23, 61-69.
See Also
pEllipticInt
Examples
p1 <- c(3,1)p2 <- c(0,2)
# Ellipse with semi-axes: a = 5, b= 2saxes <- c(5,2)
# 1 iterationarcLengthEllipse(p1,p2,saxes,n=1)
# 5 iterationsarcLengthEllipse(p1,p2,saxes,n=5)
# 10 iterationsarcLengthEllipse(p1,p2,saxes,n=10)
8 colinear
colinear Test for colinearity
Description
Tests if three points are colinear. The coordinates of the points have to be in homogeneous coordi-nates.
Usage
colinear(p1, p2, p3)
Arguments
p1 (3× 1) vector of the homogeneous coordinates of point 1.
p2 (3× 1) vector of the homogeneous coordinates of point 2.
p3 (3× 1) vector of the homogeneous coordinates of point 3.
Value
TRUE if the three points are colinear, else FALSE.
References
Richter-Gebert, Jürgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Realand Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4
Examples
# points: homogeneous coordinatesp1 <- c(3,1,1)p2 <- c(0,2,1)p3 <- c(1.5,-2,1)p4 <- c(1,3,1)
# homogeneous line passing through p1 and p2l1 <- join(p1,p2)
# homogeneous line passing through p3 and p3l2 <- join(p3,p4)
# homogeneous points formed by the intersection of the linesp5 <- meet(l1,l2)
# test for colinearitycolinear(p1, p2, p3)colinear(p1, p2, p5)colinear(p3, p4, p5)
conicMatrixToEllipse 9
# plotplot(rbind(p1,p2,p3,p4),xlim=c(-5,5),ylim=c(-5,5),asp=1)abline(h=0,v=0,col="grey",lty=3)addLine(l1,col="red")addLine(l2,col="blue")points(t(p5),cex=1.5,pch=20,col="blue")
conicMatrixToEllipse Transformation of the matrix representation of an ellipse into the el-lipse parameters
Description
Ellipses can be represented by a (3×3) matrix A, such that for each point x on the ellipse xTAx =0. The function conicMatrixToEllipse transforms the matrixA into the ellipse parameters: centerlocation, semi-axes length and angle of rotation.
Usage
conicMatrixToEllipse(A)
Arguments
A a (3× 3) matrix representation of an ellipse.
Value
loc a (2× 1) vector of the Cartesian coordinates of the ellipse center.saxes a (2× 1) vector of the length of the ellipse semi-axes.theta the angle of rotation of the ellipse (in radians).
References
Wolfram, Mathworld (http://mathworld.wolfram.com/).
See Also
ellipseToConicMatrix
Examples
# ellipse parametersaxes <- c(5,2)loc <- c(0,0)theta <- pi/4# matrix representation of the ellipseC <- ellipseToConicMatrix(saxes,loc,theta)C# back to the ellipse parametersconicMatrixToEllipse(C)
10 conicThrough5Points
conicThrough5Points Compute the conic that passes through 5 points
Description
Return the matrix representation of the conic that passes through exactly 5 points.
Usage
conicThrough5Points(p1, p2, p3, p4, p5)
Arguments
p1, p2, p3, p4, p5
(3× 1) vectors of the homogeneous coordinates of each of the five points.
Value
A (3× 3) matrix representation of the conic passing through the 5 points.
References
Richter-Gebert, Jürgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Realand Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4
Examples
# five pointsp1 <- c(-4.13, 6.24, 1)p2 <- c(-8.36, 1.17, 1)p3 <- c(-2.03, -4.61, 1)p4 <- c(9.70, -3.49, 1)p5 <- c(8.02, 3.34, 1)
# matrix representation of the conic passing# through the five pointsC5 <- conicThrough5Points(p1,p2,p3,p4,p5)
# plotplot(rbind(p1,p2,p3,p4,p5),xlim=c(-10,10), ylim=c(-10,10), asp=1)# from matrix to ellipse parametersE5 <- conicMatrixToEllipse(C5)lines(ellipse(E5$saxes, E5$loc, E5$theta, n=500))
cubic 11
cubic Roots of the cubic equation.
Description
Return the roots of a cubic equation of the form ax3 + bx2 + cx+ d = 0.
Usage
cubic(p)
Arguments
p a (4× 1) vector of the four parameters (a, b, c, d) of the cubic equation.
Value
A vector corresponding to the roots of the cubic equation.
References
W. H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery (2007). NUMERICAL RECIPES - theart of scientific computing. Cambridge, University Press, chap 5.6, p. 227-229.
Examples
# cubic equation x^3 - 6x^2 + 11x - 6 = 0# parameterb <- c(1,-6, 11, -6)
# rootsx0 <- cubic(b)
# plotx <- seq(0,4, by=0.001)y <- b[1]*x^3 + b[2]*x^2 + b[3]*x + b[4]
# plotplot(x,y,type="l")abline(h=0,v=0)points(cbind(x0,c(0,0,0)), pch=20,col="red",cex=1.8)
12 ellipse
ellipse Return ellipse points
Description
Return ellipse points. Usefull for ploting ellipses.
Usage
ellipse(saxes = c(1, 1), loc = c(0, 0), theta = 0, n = 201,method = c("default", "angle", "distance"))
Arguments
saxes a (2× 1) vector of the length of the ellipse semi-axes.
loc a (2× 1) vector of the Cartesian coordinates of the ellipse center.
theta the angle of rotation of the elllipse (in radians).
n the number of points returned by the function.
method "default" returns points according to the polar equation; "angle" returns pointsradially equidistant; "distance" returns points that are equidistant on the el-lipse arc.
Value
A (n× 2) matrix whose columns correspond to the Cartesian coordinates of the points lying on theellipse.
Examples
# Ellipse parameterssaxes <- c(5,2)loc <- c(0,0)theta <- pi/4
# Plotplot(ellipse(saxes, loc, theta, n=500),type="l")points(ellipse(saxes, loc, theta, n=30),pch=20,col="red")points(ellipse(saxes, loc, theta, n=30, method="angle"),pch=20,col="blue")points(ellipse(saxes, loc, theta, n=30, method="distance"),pch=20,col="green")
ellipseToConicMatrix 13
ellipseToConicMatrix Transformation of the ellipse parameters into the matrix representa-tion
Description
Transformation of the ellipse parameters (Cartesian coordinates of the ellipse center, length of thesemi-axes and angle of rotation) into the (3× 3) into the matrix representation of conics.
Usage
ellipseToConicMatrix(saxes = c(1, 1), loc = c(0, 0), theta = 0)
Arguments
saxes a (2× 1) vector of the length of the ellipse semi-axes.
loc a (2× 1) vector of the Cartesian coordinates of the ellipse center.
theta the angle of rotation of the ellipse (in radians).
Value
A (3× 3) matrix that represents the ellipse.
See Also
conicMatrixToEllipse
Examples
# Ellipse parameterssaxes <- c(5,2)loc <- c(0,0)theta <- pi/4# Matrix representation of the ellipseC <- ellipseToConicMatrix(saxes,loc,theta)
Intersection with conics
Intersections with conics
Description
Point(s) of intersection between a conic and a line, and between two conics in homogeneous coor-dinates.
14 Intersection with conics
Usage
intersectConicLine(C, l)intersectConicConic(C1,C2)
Arguments
C, C1, C2 (3× 3) matrix representation of conics.
l a (3× 3) vector of the homogeneous representation of a line.
Value
The homogeneous coordinates of the intersection points. If there are two points of intersection, itreturns a (3 × 2) matrix whose columns correspond to the homogeneous coordinates of the inter-section points. If there is only one point, a (3 × 1) vector of the homogeneous coordinates of theintersection point is returned. If there is no intersection, NULL is returned.
References
Richter-Gebert, Jürgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Realand Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4
Examples
# Ellipse with semi-axes a=8, b=2, centered in (0,0), with orientation angle = -pi/3C <- ellipseToConicMatrix(c(8,2),c(0,0),-pi/3)
# Ellipse with semi-axes a=5, b=2, centered in (1,-2), with orientation angle = pi/5C2 <- ellipseToConicMatrix(c(5,2),c(1,-2),pi/5)
# linel <- c(0.25,0.85,-3)
# intersection conic C with line l:p_Cl <- intersectConicLine(C,l)
# intersection conic C with conic C2p_CC2 <- intersectConicConic(C,C2)
# plotplot(ellipse(c(8,2),c(0,0),-pi/3),type="l",asp=1)lines(ellipse(c(5,2),c(1,-2),pi/5), col="blue")addLine(l,col="red")points(t(p_Cl), pch=20,col="red")points(t(p_CC2), pch=20,col="blue")
Join, meet and parallel 15
Join, meet and parallel
The join and meet of two points and the parallel
Description
The join operation of two points is the cross-product of these two points and represents the linepassing through them. The meet operation of two lines is the cross-product of these two lines andrepresents their intersection. The line parallel to a line l and passing through the point p correspondsto the join of p with the meet of l and the line at infinity.
Usage
join(p,q)meet(l, m)parallel(p,l)
Arguments
p, q (3× 1) vectors of the homogeneous coordinates of a point.
l, m (3× 1) vectors of the homogeneous representation of a line.
Value
A (3 × 1) vector of either the homogeneous coordinates of the meet of two lines (a point), thehomogeneous representation of the join of two points (line), or the homogeneous representation ofthe parallel line. The vector has the form (x, y, 1).
References
Richter-Gebert, Jürgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Realand Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4
Examples
p <- c(3,1,1)q <- c(0,2,1)l <- c(0.75,0.25,1)
# m is the line passin through p and qm <- join(p,q)
# intersection point of m and lml <- meet(l,m)
# line parallel to l and through plp <- parallel(p,l)
16 pEllipticInt
# plotplot(rbind(p,q),xlim=c(-5,5),ylim=c(-5,5))abline(h=0,v=0,col="grey",lty=3)addLine(l,col="red")addLine(m,col="blue")points(t(ml),cex=1.5,pch=20,col="blue")addLine(lp,col="green")
pEllipticInt Partial elliptic integral
Description
Partial elliptic integral
Usage
pEllipticInt(x, saxes, n = 5)
Arguments
x the x-coordinate.saxes a (2× 1) vector of the length of the ellipse semi-axes.n the number of iterations.
Value
Return the partial elliptic integral.
References
Van de Vel, H. (1969). On the series expansion method for Computing incomplete elliptic integralsof the first and second kinds, Math. Comp. 23, 61-69.
See Also
arcLengthEllipse
Examples
# Ellipse with semi-axes: a = 5, b= 2saxes <- c(5,2)
# 1 iterationpEllipticInt(3,saxes,n=1)# 5 iterationspEllipticInt(3,saxes,n=5)# 10 iterationspEllipticInt(3,saxes,n=10)
polar 17
polar Polar line of point with respect to a conic
Description
Return the polar line l of a point p with respect to a conic with matrix representation C. The polarline l is defined by l = Cp.
Usage
polar(p, C)
Arguments
p a (3× 1) vector of the homogeneous coordinates of a point.
C a (3× 3) matrix representation of the conic.
Details
The polar line of a point p on a conic is tangent to the conic on p.
Value
A (3× 1) vector of the homogeneous representation of the polar line.
References
Richter-Gebert, Jürgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Realand Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4
Examples
# Ellipse with semi-axes a=5, b=2, centered in (1,-2), with orientation angle = pi/5C <- ellipseToConicMatrix(c(5,2),c(1,-2),pi/5)
# linel <- c(0.25,0.85,-1)
# intersection conic C with line l:p_Cl <- intersectConicLine(C,l)
# if p is on the conic, the polar line is tangent to the conicl_p <- polar(p_Cl[,1],C)
# point outside the conicp1 <- c(5,-3,1)l_p1 <- polar(p1,C)
# point inside the conic
18 quadraticFormToMatrix
p2 <- c(-1,-4,1)l_p2 <- polar(p2,C)
# plotplot(ellipse(c(5,2),c(1,-2),pi/5),type="l",asp=1, ylim=c(-10,2))# addLine(l,col="red")points(t(p_Cl[,1]), pch=20,col="red")addLine(l_p,col="red")points(t(p1), pch=20,col="blue")addLine(l_p1,col="blue")points(t(p2), pch=20,col="green")addLine(l_p2,col="green")
# DUAL CONICSsaxes <- c(5,2)theta <- pi/7E <- ellipse(saxes,theta=theta, n=50)C <- ellipseToConicMatrix(saxes,c(0,0),theta)plot(E,type="n",xlab="x", ylab="y", asp=1)points(E,pch=20)E <- rbind(t(E),rep(1,nrow(E)))
All_tangant <- polar(E,C)apply(All_tangant, 2, addLine, col="blue")
quadraticFormToMatrix Transformation of the quadratic conic representation into the matrixrepresentation.
Description
Transformation of the quadratic conic representation into the matrix representation.
Usage
quadraticFormToMatrix(v)
Arguments
v a (6 × 1) vector of the parameters (a, b, c, d, e, f) of the quadratic form ax2 +bxy + cy2 + dxz + eyz + fz2 = 0.
Value
A (3× 3) matrix representation of the conic (symmetric matrix).
Examples
v <- c(2,2,-2,-20,20,10)quadraticFormToMatrix(v)
skewSymmetricMatrix 19
skewSymmetricMatrix (3× 3) skew symmetric matrix
Description
Return a (3× 3) skew symmetric matrix from three parameters (λ, µ, τ).
Usage
skewSymmetricMatrix(p)
Arguments
p a (3× 1) vector (λ, µ, τ).
Value
A (3× 3) skew symmetric matrix, with :
• A1,2 = −A2,1 = τ
• −A1,3 = A3,1 = µ
• A3,2 = −A2,3 = λ
References
Richter-Gebert, Jürgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Realand Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4
Examples
p <- c(3,7,11)skewSymmetricMatrix(p)
splitDegenerateConic Split degenerate conic
Description
Split a degenerate conic into two lines.
Usage
splitDegenerateConic(C)
20 splitDegenerateConic
Arguments
C a (3× 3) matrix representation of a degenerate conic.
Value
A (3× 2) matrix whose columns correspond to the homongeneous representation of two lines (realor complex).
References
Richter-Gebert, Jürgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Realand Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4
Examples
# tw0 linesg <- c(0.75,0.25,3)h <- c(0.5,-0.25,2)
# a degenerate conicD <- g %*% t(h) + h %*% t(g)
# split the degenerate conic into 2 linesL <- splitDegenerateConic(D)
# plotplot(0,0,xlim=c(-10,5),ylim=c(-10,10),type="n")addLine(L[,1],col="red")addLine(L[,2],col="green")
Index
∗Topic conicRConics-package, 2
(i,j)-cofactor and (i,j)-minor, 3
addLine, 4adjoint, 3, 4Affine planar transformations matrix, 5affine planar transformations matrix
(Affine planar transformationsmatrix), 5
arcLengthEllipse, 6, 16
cofactor, 5cofactor ((i,j)-cofactor and
(i,j)-minor), 3colinear, 8conicMatrixToEllipse, 9, 13conicThrough5Points, 10cubic, 11
ellipse, 12ellipseToConicMatrix, 9, 13
graphical parameters, 4
intersectConicConic (Intersection withconics), 13
intersectConicLine (Intersection withconics), 13
Intersection with conics, 13
join (Join, meet and parallel), 15Join, meet and parallel, 15
meet (Join, meet and parallel), 15minor, 5minor ((i,j)-cofactor and (i,j)-minor),
3
parallel (Join, meet and parallel), 15pEllipticInt, 7, 16
polar, 17
quadraticFormToMatrix, 18
RConics (RConics-package), 2RConics-package, 2reflection (Affine planar
transformations matrix), 5rotation (Affine planar
transformations matrix), 5
scaling (Affine planar transformationsmatrix), 5
skewSymmetricMatrix, 19splitDegenerateConic, 19
translation (Affine planartransformations matrix), 5
21